As far as I can tell, the word “‍random‍” is not really defined
in most people’s minds.
When I press for a definition I very rarely get anything meaningful;
when I do, it is often a definition
that doesn’t match the context that inspired me to seek clarity.
Here I try to delineate some of these definitions,
as well as some others I find interesting.

Not Formulaic

Some things follow mathematically-precise models.
Everything else is “‍random‍”.
Lightning was random until we discovered static.
Usually people add in a speculative allowance for unknown models
so that they don’t have to say anything changes from random to non-random.
Of course, that also means that we have no idea if any particular thing is random or not.

Quantum physics gives an even stronger version of this,
demonstrating that the existence of a wide class of models of quantum wave collapse
are inconsistent with quantum field theory.
Of course, if we found one of those models did exist
all we’d have done is invalidated quantum field theory;
see my post on confusing what works for what is.

Unpredictable

If you knew absolutely everything that exists
at the present and all past times,
that which you could not predict is random.
This merges with the previous definition
if you assume that the only means of prediction is formulaic reason.
Otherwise, it is even stronger,
excluding free will (because you could knowing people)
and prophesy, to name just two.

This is always speculative;
labeling anything this kind of random is an act of pure faith
or remarkable audacity.

Unrevealing

A dual of being unpredictable is being unrevealing.
If adding the knowledge of an event cannot increase our understanding
of any prior event, no matter what base knowledge pool we add it to,
then the event was random.

Unrevealing is very close to being unpredictable.
An oracle that often says true things about the past
is not unrevealing-random but is unpredictable-random.
An oracle that often says true things about otherwise-unpredictable future events
is not unpredictable-random but is unrevealing-random.

Meaningless

This subsumes and extends unpredictable and unrevealing.
If, given the entire universe over all time,
the only patterns that can be stated about the event
are one where it was the cause of other events,
then it is random.

The spontaneous uncaused unpredicted unpredictable appearance of a hippopotamus
with no correlation to any prior event
is meaningless-random.

Chaotic

A chaotic system is one that is arbitrarily sensitive to initial conditions.
Something that enumerates all of the digits of a real number is chaotic;
any change to the number, no matter how small,
will eventually completely change the output.
The weather and strange attractors are other well-known chaotic systems.

Chaotic-random means simply
“‍too complicated to have figured out in advance.‍”
Die rolls are this kind of random.
I might meet a random person on my way home today,
or randomly sneeze.
Chaos is the only kind of random I really understand.

My many ideas of what random might mean aside,
I still usually have no idea what people mean when they refer to it.
I have a lingering doubt that there is some kind of random that I don’t understand,
or some reason why all the non-chaos randoms must actually be the same thing.